3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
A mirror and a source of light are situated at the origin $O$ and at a point on the line $OX$ respectively. A ray of light from the source strikes the mirror at $O$ and is reflected. If the direction ratios of the normal to the plane of the mirror are $(1, -1, 1)$; then the direction cosines of the reflected ray are :
$\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$
$\frac{-1}{3}, \frac{2}{3}, \frac{2}{3}$
$\frac{-1}{3}, \frac{-2}{3}, \frac{2}{3}$
$\frac{-1}{3}, \frac{-2}{3}, \frac{2}{3}$
Step-by-Step Solution
Key Concept: The normal to the mirror bisects the angle between incident and reflected rays; use the unit vector condition to solve.
For incident ray with direction $(-1, 0, 0)$ and reflected ray direction $(l, m, n)$, the normal direction satisfies $\frac{l+1}{1} = \frac{m}{-1} = \frac{n}{1} = k$. Using the unit normal constraint $l^2 + m^2 + n^2 = 1$, we get $(k-1)^2 + (-k)^2 + k^2 = 1$, yielding $k = \frac{2}{3}$. Thus the direction of the reflected ray is $\left(-\frac{1}{3}, -\frac{2}{3}, \frac{2}{3}\right)$.
Correct Answer: 3